Problem: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{-10p^3 - 30p^2 + 540p}{-9p^3 - 126p^2 - 405p}$
First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {-10p(p^2 + 3p - 54)} {-9p(p^2 + 14p + 45)} $ $ y = \dfrac{10p}{9p} \cdot \dfrac{p^2 + 3p - 54}{p^2 + 14p + 45} $ Simplify: $ y = \dfrac{10}{9} \cdot \dfrac{p^2 + 3p - 54}{p^2 + 14p + 45}$ Since we are dividing by $p$ , we must remember that $p \neq 0$ Next factor the numerator and denominator. $ y = \dfrac{10}{9} \cdot \dfrac{(p + 9)(p - 6)}{(p + 9)(p + 5)}$ Assuming $p \neq -9$ , we can cancel the $p + 9$ $ y = \dfrac{10}{9} \cdot \dfrac{p - 6}{p + 5}$ Therefore: $ y = \dfrac{ 10(p - 6)}{ 9(p + 5)}$, $p \neq -9$, $p \neq 0$